Question

The Tire Rack, America’s leading online distributor of tires and wheels, conducts extensive testing to provide customers with products that are right for their vehicle, driving style, and driving conditions. In addition, the Tire Rack maintains an independent consumer survey to help drivers help each other by sharing their long-term tire experiences. The following data show survey ratings (1 to 10 scale with 10 the highest rating) for 18 maximum performance summer tires. The variable Steering rates the tire’s steering responsiveness, Tread Wear rates quickness of wear based on the driver’s expectations, and Buy Again rates the driver’s overall tire satisfaction and desire to purchase the same tire again. Use Excel Formula(Excel only)

Tire	Steering	Tread Wear	Buy Again
Goodyear Assurance TripleTred	8.9	8.5	8.1
Michelin HydroEdge	8.9	9.0	8.3
Michelin Harmony	8.3	8.8	8.2
Dunlop SP 60	8.2	8.5	7.9
Goodyear Assurance ComforTred	7.9	7.7	7.1
Yokohama Y372	8.4	8.2	8.9
Yokohama Aegis LS4	7.9	7.0	7.1
Kumho Power Star 758	7.9	7.9	8.3
Goodyear Assurance	7.6	5.8	4.5
Hankook H406	7.8	6.8	6.2
Michelin Energy LX4	7.4	5.7	4.8
Michelin MX4	7.0	6.5	5.3
Michelin Symmetry	6.9	5.7	4.2
Kumho 722	7.2	6.6	5.0
Dunlop SP 40 A/S	6.2	4.2	3.4
Bridgestone Insignia SE200	5.7	5.5	3.6
Goodyear Integrity	5.7	5.4	2.9
Dunlop SP20 FE	5.7	5.0	3.3

1. At the .05 level, are there significant relationship between the dependent variable and the independent variables?
2. What is the multiple coefficient of determination?
3. What is the adjusted multiple coefficient of determination?
4. Interpret the results.

Answer 1

Buy again is the dependent variable and other two are the independent variables.

We study regression of 'Buy again' on other three or we use other three to estimate 'Repair Time'.

a. At the .05 level, are there significant relationship between the dependent variable and the independent variables?

Ho: The model is not significant or b0 = b1 = b2 = 0

H1: The model is significant or atleast 1 bi is not equal to 0.

For this, we will run multiple linear regression and will apply ANOVA test to see if the relationship is significant.

Steps in Excel:

Input Data > Data > Data Analysis > Regression > Input 'Repair Time' as Y > Input 'other 2' as X > OK

Data (this should be your data):

Tire	Steering	Tread Wear	Buy Again
Goodyear Assurance TripleTred	8.9	8.5	8.1
Michelin HydroEdge	8.9	9	8.3
Michelin Harmony	8.3	8.8	8.2
Dunlop SP 60	8.2	8.5	7.9
Goodyear Assurance ComforTred	7.9	7.7	7.1
Yokohama Y372	8.4	8.2	8.9
Yokohama Aegis LS4	7.9	7	7.1
Kumho Power Star 758	7.9	7.9	8.3
Goodyear Assurance	7.6	5.8	4.5
Hankook H406	7.8	6.8	6.2
Michelin Energy LX4	7.4	5.7	4.8
Michelin MX4	7	6.5	5.3
Michelin Symmetry	6.9	5.7	4.2
Kumho 722	7.2	6.6	5
Dunlop SP 40 A/S	6.2	4.2	3.4
Bridgestone Insignia SE200	5.7	5.5	3.6
Goodyear Integrity	5.7	5.4	2.9
Dunlop SP20 FE	5.7	5	3.3

Output:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.965279	Part b
R Square	0.931764	Part c
Adjusted R Square	0.922665
Standard Error	0.572723
Observations	18
ANOVA
	df	SS	MS	F	Significance F
Regression	2	67.18482	33.59241	102.4121	1.79936E-09
Residual	15	4.920181	0.328012
Total	17	72.105
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-5.38767	1.109533	-4.8558	0.00021	-7.752587095	-3.022758286
X Variable 1	0.689871	0.287547	2.399155	0.029874	0.07697806	1.302763926
X Variable 2	0.91133	0.206343	4.416569	0.0005	0.471519514	1.351140807

F = 102.4121 and p-value = 1.799*10^-9 < 0.05, so we reject Ho or can say that the Model or this regression is significant.

b. What is the multiple coefficient of determination?

R Square

0.9318

c. What is the adjusted multiple coefficient of determination?

Adjusted R Square

0.9227

d. Interpret the results.

• F = 102.4121 and p-value = 1.799*10^-9 < 0.05, so we reject Ho or can say that the Model or this regression is significant.
• R-square = 0.9318 or the model explains 93.18% variability in 'Repair Time'
• Model Equation: Buy Again = -5.39 + 0.69 Steering + 0.91 Tread Wear
• From p-value of each variable, all are significant as p-value < 0.05.

Please rate my answer and comment for doubt.